Notice now that, due to \(\left(\small \begin{bmatrix}C_{12}&C_{13}\end{bmatrix},\footnotesize \begin{bmatrix}A_{22}&A_{23}\\O&A_{33}\end{bmatrix}\right)\)  being observable and \(A_{11}-\lambda I_{n_{11}}\)  being invertible \(\forall \lambda \in \mathbb {C}\backslash \mathbb {S}\) , the realization from (REF ) is detectable and, due to \((A_{22},B_{21})\)  being controllable and both \(A_{11}-\lambda I_{n_{11}}\)  along with \(A_{33}-\lambda I_{n_{33}}\)  being invertible \(\forall \lambda \in \mathbb {C}\backslash \mathbb {S}\) , the realization from (REF ) is also stabilizable. By Theorem 3.5 of [1]}, we also conclude that the realization from (REF ) is also detectable and stabilizable.
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